# Combination Theorem for Limits of Functions/Product Rule

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## Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $f$ and $g$ be functions defined on an open subset $S \subseteq X$, except possibly at the point $c \in S$.

Let $f$ and $g$ tend to the following limits:

- $\displaystyle \lim_{x \mathop \to c} \map f x = l$
- $\displaystyle \lim_{x \mathop \to c} \map g x = m$

Then:

- $\displaystyle \lim_{x \mathop \to c} \ \paren {\map f x \map g x} = l m$

## Proof

Let $\sequence {x_n}$ be any sequence of points of $S$ such that:

- $\forall n \in \N: x_n \ne c$
- $\displaystyle \lim_{n \mathop \to \infty} x_n = c$

By Limit of Function by Convergent Sequences:

- $\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = l$
- $\displaystyle \lim_{n \mathop \to \infty} \map g {x_n} = m$

By the Product Rule for Sequences:

- $\displaystyle \lim_{n \mathop \to \infty} \ \paren {\map f {x_n} \map g {x_n} } = l m$

Applying Limit of Function by Convergent Sequences again, we get:

- $\displaystyle \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$

$\blacksquare$

## Sources

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(ii)}$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 8.12 \ \text{(ii)}$